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General upper tail estimates are given for counting edges in a random induced subhypergraph of a fixed hypergraph ℋ, with an easy proof by estimating the moments. As an application we consider the numbers of arithmetic progressions and Schur triples in random subsets of integers. In the second part of the paper we return to the subgraph counts in random graphs and provide upper tail estimates in the rooted case.

In a uniform random recursive$k$-directed acyclic graph, there is a root, 0, and each node in turn, from 1 to$n$, chooses$k$uniform random parents from among the nodes of smaller index. If$S$_{$n$}is the shortest path distance from node$n$to the root, then we determine the constant$σ$such that$S$_{$n$}/log$n$→$σ$in probability as$n$→∞. We also show that max_{1≤$i$≤$n$}$S$_{$i$}/log$n$→$σ$in probability.

Let each point of a homogeneous Poisson process in ℝ^{$d$}independently be equipped with a random number of stubs (half-edges) according to a given probability distribution$μ$on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution$μ$. Leaving aside degenerate cases, we prove that for any$μ$there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme which is a natural extension of Gale–Shapley stable marriage, we give sufficient conditions on$μ$for the absence and presence of infinite components.

It is well known that complementarity functions play an important role in dealing with complementarity problems. In this paper, we propose a few new classes of complementarity functions for nonlinear complementarity problems and second-order cone complementarity problems. The constructions of such new complementarity functions are based on discrete generalization which is a novel idea in contrast to the continuous generalization of FischerBurmeister function. Surprisingly, these new families of complementarity functions possess continuous differentiability even though they are discrete-oriented extensions. This feature enables that some methods like derivative-free algorithm can be employed directly for solving nonlinear complementarity problems and second-order cone complementarity problems. This is a new discovery to the literature and we believe that such new complementarity functions